p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.261C23, (C4×D4).24C4, (C4×Q8).23C4, C4⋊C8.230C22, (C4×M4(2))⋊32C2, (C2×C4).642C24, (C4×C8).326C22, (C2×C8).399C23, C42.205(C2×C4), C42.6C4⋊44C2, C8⋊C4.153C22, C4.17(C42⋊C2), C2.11(Q8○M4(2)), C22⋊C8.138C22, C23.101(C22×C4), (C22×C8).430C22, (C2×C42).755C22, C22.170(C23×C4), (C22×C4).913C23, C42.7C22⋊20C2, C42.6C22⋊28C2, C22.5(C42⋊C2), C42⋊C2.291C22, (C2×M4(2)).344C22, (C2×C8⋊C4)⋊32C2, C4⋊C4.218(C2×C4), (C4×C4○D4).12C2, C4.293(C2×C4○D4), (C2×D4).228(C2×C4), C22⋊C4.69(C2×C4), (C2×Q8).206(C2×C4), (C2×C4).680(C4○D4), (C2×C4).258(C22×C4), (C22×C4).336(C2×C4), C2.42(C2×C42⋊C2), (C22×C8)⋊C2.18C2, (C2×C4○D4).281C22, SmallGroup(128,1655)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 268 in 192 conjugacy classes, 132 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×2], C2×C4 [×14], C2×C4 [×10], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×2], C42 [×8], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×8], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×6], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42, C2×C42 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×C4○D4, C2×C8⋊C4, C4×M4(2), (C22×C8)⋊C2 [×2], C42.6C22 [×2], C42.6C4 [×4], C42.7C22 [×4], C4×C4○D4, C42.261C23
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, Q8○M4(2) [×2], C42.261C23
Generators and relations
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, ad=da, eae=ab2, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, ede=b2d >
►On 64 points
(1 33 55 43)(2 48 56 38)(3 35 49 45)(4 42 50 40)(5 37 51 47)(6 44 52 34)(7 39 53 41)(8 46 54 36)(9 26 64 20)(10 17 57 31)(11 28 58 22)(12 19 59 25)(13 30 60 24)(14 21 61 27)(15 32 62 18)(16 23 63 29)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 23)(2 30)(3 17)(4 32)(5 19)(6 26)(7 21)(8 28)(9 34)(10 45)(11 36)(12 47)(13 38)(14 41)(15 40)(16 43)(18 50)(20 52)(22 54)(24 56)(25 51)(27 53)(29 55)(31 49)(33 63)(35 57)(37 59)(39 61)(42 62)(44 64)(46 58)(48 60)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)
G:=sub<Sym(64)| (1,33,55,43)(2,48,56,38)(3,35,49,45)(4,42,50,40)(5,37,51,47)(6,44,52,34)(7,39,53,41)(8,46,54,36)(9,26,64,20)(10,17,57,31)(11,28,58,22)(12,19,59,25)(13,30,60,24)(14,21,61,27)(15,32,62,18)(16,23,63,29), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,23)(2,30)(3,17)(4,32)(5,19)(6,26)(7,21)(8,28)(9,34)(10,45)(11,36)(12,47)(13,38)(14,41)(15,40)(16,43)(18,50)(20,52)(22,54)(24,56)(25,51)(27,53)(29,55)(31,49)(33,63)(35,57)(37,59)(39,61)(42,62)(44,64)(46,58)(48,60), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60)>;
G:=Group( (1,33,55,43)(2,48,56,38)(3,35,49,45)(4,42,50,40)(5,37,51,47)(6,44,52,34)(7,39,53,41)(8,46,54,36)(9,26,64,20)(10,17,57,31)(11,28,58,22)(12,19,59,25)(13,30,60,24)(14,21,61,27)(15,32,62,18)(16,23,63,29), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,23)(2,30)(3,17)(4,32)(5,19)(6,26)(7,21)(8,28)(9,34)(10,45)(11,36)(12,47)(13,38)(14,41)(15,40)(16,43)(18,50)(20,52)(22,54)(24,56)(25,51)(27,53)(29,55)(31,49)(33,63)(35,57)(37,59)(39,61)(42,62)(44,64)(46,58)(48,60), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,61)(34,62)(35,63)(36,64)(37,57)(38,58)(39,59)(40,60) );
G=PermutationGroup([(1,33,55,43),(2,48,56,38),(3,35,49,45),(4,42,50,40),(5,37,51,47),(6,44,52,34),(7,39,53,41),(8,46,54,36),(9,26,64,20),(10,17,57,31),(11,28,58,22),(12,19,59,25),(13,30,60,24),(14,21,61,27),(15,32,62,18),(16,23,63,29)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,23),(2,30),(3,17),(4,32),(5,19),(6,26),(7,21),(8,28),(9,34),(10,45),(11,36),(12,47),(13,38),(14,41),(15,40),(16,43),(18,50),(20,52),(22,54),(24,56),(25,51),(27,53),(29,55),(31,49),(33,63),(35,57),(37,59),(39,61),(42,62),(44,64),(46,58),(48,60)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,61),(34,62),(35,63),(36,64),(37,57),(38,58),(39,59),(40,60)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 6 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 14 | 13 | 0 | 0 | 0 | 0 |
| 11 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 7 |
| 0 | 0 | 10 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 | 0 | 7 |
| 0 | 0 | 7 | 0 | 7 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 10 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(17))| [13,6,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,13,0,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[14,11,0,0,0,0,13,3,0,0,0,0,0,0,0,10,0,7,0,0,10,0,7,0,0,0,0,7,0,7,0,0,7,0,7,0],[16,10,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T | 8A | ··· | 8P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4○D4 | Q8○M4(2) |
| kernel | C42.261C23 | C2×C8⋊C4 | C4×M4(2) | (C22×C8)⋊C2 | C42.6C22 | C42.6C4 | C42.7C22 | C4×C4○D4 | C4×D4 | C4×Q8 | C2×C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 12 | 4 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{261}C_2^3 % in TeX
G:=Group("C4^2.261C2^3"); // GroupNames label
G:=SmallGroup(128,1655);
// by ID
G=gap.SmallGroup(128,1655);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,100,1018,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations